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Theorem eldmcoss 36852
Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
Assertion
Ref Expression
eldmcoss (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem eldmcoss
StepHypRef Expression
1 dmcoss3 36847 . . 3 dom ≀ 𝑅 = dom 𝑅
21eleq2i 2830 . 2 (𝐴 ∈ dom ≀ 𝑅𝐴 ∈ dom 𝑅)
3 eldmcnv 36738 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
42, 3bitrid 283 1 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1782  wcel 2107   class class class wbr 5104  ccnv 5631  dom cdm 5632  ccoss 36566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-coss 36805
This theorem is referenced by:  eldmcoss2  36853  eldm1cossres  36854
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